Retracted Article: Strong convergence theorem by a new iterative method for equilibrium problems and symmetric generalized hybrid mappings

Abstract

In this paper, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping by using modified hybrid method. Our results extend and improve some existing results in the literature.

Shrinking Extragradient Method for Pseudomonotone Equilibrium Problems and Quasi-Nonexpansive Mappings

“…On the other hand, for a nonempty closed convex subset C of H and a mapping T : C → C, the fixed point problem is a problem of finding a point x ∈ C such that Tx = x. This fixed point problem has many important applications, such as optimization problems, variational inequality problems, minimax problems, and saddle point problems, see [8][9][10][11], and the references therein. The set of fixed points of a mapping T will be represented by Fix(T).…”

“…In recent years, many algorithms have been proposed for finding a common element of the set of solutions of the equilibrium problem and the set of solutions of the fixed point problem. See, for instance, [8,11,[18][19][20][21][22][23] and the references therein. In 2016, by using both hybrid and extragradient methods together in combination with Ishikawa's iteration concept, Dinh and Kim [24] proposed the following iteration method for finding a common element of fixed points of a symmetric generalized hybrid mapping T and the set of solutions of the equilibrium problem, when a bifunction f is pseudomonotone and Lipschitz-type continuous with positive constants L 1 and L 2 :…”

Weak convergence theorems for a symmetric generalized hybrid mapping and an equilibrium problem

“…Motivated by these facts and recent works [6,22,36], in this paper, we combine Ishikawa's algorithm with solution methods for equilibrium problems for finding a common element of the set of fixed points of a generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space in which the mapping T is symmetric generalized hybrid, and the bifunction f is monotone on C or pseudomonotone on C with respect to its solution set. More precisely, we propose to use the Ishikawa's algorithm for finding a fixed point of the mapping T by incorporating it with the proximal point algorithm and the extragradient algorithms with or without linesearch [20] for solving the equilibrium problem EP(C, f ) (see also [7,8,10,19,32] for more details on the extragradient algorithms).…”